Regularity of a boundary having a Schwarz function. (English) Zbl 0728.30007

Summary: In his book, P. J. Davis discussed various interesting aspects concerning a Schwarz function. It is a holomorpic function S defined in a neighbourhood of a real analytic arc satisfying \(S(\zeta)={\bar \zeta}\) on the arc, where \({\bar \zeta}\) denotes the complex conjugate of \(\zeta\). The author defines a Schwarz function for a portion of a boundary of an arbitrary open set and shows the regularity of the portion of the boundary. More precisely, let \(\Omega\) be an open set of the unit disk B such that the boundary \(\partial \Omega\) contains the origin 0 and let \(\Gamma =(\partial \Omega)\cap B\). A function S defined on \(\Omega\cup \Gamma\) is called the Schwarz function of \(\Omega\cup \Gamma\) if (i) S is holomorphic in \(\Omega\), (ii) S is continuous on \(\Omega\cup \Gamma\) and (iii) \(S(\zeta)={\bar \zeta}\) on \(\Gamma\). The author gives a classification of a boundary having a Schwarz function. The main theorem asserts that there are four types of the boundary if 0 is not an isolated boundary point of \(\Omega\) : 0 is a regular, nonisolated degenerate, double or cusp point of the boundary.


30C35 General theory of conformal mappings
30C99 Geometric function theory
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